Newton slopes for Artin-Schreier-Witt towers

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.Comment: 15 pages, upon the refereed version (to appear in Math. Ann), we fixed two minor errors, one in the proof of Theorem 3.8, the other for Theorem 4.

Slopes for higher rank Artin-Schreier-Witt Towers

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower. This extends the main result in [DWX] from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$.Comment: 20 page

Robust Half-Metallic Character and Large Oxygen Magnetism in a Perovskite Cuprate

The new perovskite cuprate material Sr$_{8}$CaRe$_{3}$Cu$_{4}$O$_{24}$, which behaves ferrimagnetically and shows an unusually high Curie temperature ($T_c \sim$ 440 K), is found from density-functional theory calculation to display several surprising properties after hole doping or chemical substitution: (1) Half metal (HM) is realized by replacing Re with W or Mo while $T_c$ remains high; (2) hole-doped Sr$_{8}$CaRe$_{3}$Cu$_{4}$O$_{24}$ is also HM with high $T_c$. Moreover, we find that the O atoms will carry a large magnetic moment after hole doping, which is in sharp contrast with the generally accepted concept that magnetism in solid requires partially filled shells of $d$ or $f$ electrons in cations. The material Sr$_8$CaRe$_3$Cu$_4$O$_{24}$ is therefore expected to provide a very useful platform for material design and development.Comment: 5 pages and 4 figure